We begin by stating the standard guarantee for the Perceptron algorithm due to [1]}. To emphasizes its generality, we use \(\bar{x}_i\)  to denote its inputs, which we allow to have non-normalized radius \(R\) .
[!t]
[1]
Initialize  \(w_1 = \mathbf {e_1} \in \mathcal {B}_1^d\) 
\(t=1,2,\dots \) 
Recieve  \(x_t\)  and predict
\(\hat{y}_t = \operatorname{sign}(\left\langle w_t , x_t \right\rangle ), \) 
